Abstract
We consider the effective theory of perturbative quantum gravity coupled to a point particle, quantizing fluctuations of both the gravitational field and the particle’s position around flat space. Using a recent relational approach to construct gauge-invariant observables, we compute one-loop graviton corrections to the invariant metric perturbation, whose time-time component gives the Newtonian gravitational potential. The resulting quantum correction consists of two parts: the first stems from graviton loops and agrees with the correction derived by other methods, while the second one is sourced by the quantum fluctuations of the particle’s position and energy-momentum, and may be viewed as an analog of a “Zitterbewegung”. As a check on the computation, we also recover classical corrections which agree with the perturbative expansion of the Schwarzschild metric.
Highlights
For gravitons it depends on the method that is used to compute the corrections and define the potential
To determine a gauge-invariant observable corresponding to the Newtonian potential, we follow instead the recent proposal [44] and its extensions [45,46,47,48], where relational observables for the gravitational field were constructed to all orders in perturbation theory, and which at linear order reduce to the Bardeen variables
Summarising, the aim of the calculation that we present here is the computation of graviton loop corrections to the Newtonian potential by a new method: instead of using the inverse scattering method, we compute the expectation value of an invariant observable corresponding to the metric perturbation, sourced by a point particle
Summary
Following refs. [44,45,46,47,48], we construct gauge-invariant observables in a relational way. One needs to define field-dependent coordinates that transform as scalars under diffeomorphisms of the background spacetime coordinates. The invariant observable corresponding to the metric perturbation is defined by performing a diffeomorphism of the metric gμν to the new coordinates X(α) and subtracting the flat background metric: Hμν (X). Using the transformation of the metric perturbation (2.2) and the coordinate corrections (2.11) under diffeomorphisms, we obtain δξHμν = 2∂(μξν) − ηρμ∂ν ξρ − ηρν ∂μξρ + O(κ) = O(κ) ,. We will choose the exact (Landau-type) gauge J(μ1)[h] = 0 in subsection 2.3, and can neglect all terms involving J(μ1) in the expansion This is a significant simplification, and up to second order we only need.
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